Category:Definitions/Total Variation of Real Function
This category contains definitions related to Total Variation of Real Function in the context of Real Analysis.
Related results can be found in Category:Total Variation of Real Function.
Closed Bounded Interval
Let $a, b$ be real numbers with $a < b$.
Let $f: \closedint a b \to \R$ be a function of bounded variation.
Let $\map X {\closedint a b}$ be the set of finite subdivisions of $\closedint a b$.
For each $P \in \map X {\closedint a b}$, write:
- $P = \set {x_0, x_1, \ldots, x_n}$
with:
- $a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
Also write:
- $\ds \map {V_f} {P ; \closedint a b} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$
We define the total variation $\map {V_f} {\closedint a b}$ of $f$ on $\closedint a b$ by:
- $\ds \map {V_f} {\closedint a b} = \map {\sup_{P \mathop \in \map X {\closedint a b} } } {\map {V_f} {P ; \closedint a b} }$
Closed Unbounded Interval
Let $I$ be an unbounded closed interval or $\R$.
Let $f: I \to \R$ be a real function.
Let $\map {\PP_F} I$ be the set of finite subsets of $I$.
For each finite non-empty subset $\SS$ of $I$, write:
- $\SS = \set {x_0, x_1, \ldots, x_n}$
with:
- $x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n$
Also write:
- $\ds \map {V_f^\ast} {\SS; I} = \sum_{i \mathop = 1}^n \size {\map f {x_i} - \map f {x_{i - 1} } }$
We define the total variation $\map {V_f} I$ of $f$ on $I$ by:
- $\ds \map {V_f} I = \sup_{\SS \mathop \in \map {\PP_F} I} \paren {\map {V_f^\ast} {\SS; I} }$
Subcategories
This category has only the following subcategory.
Pages in category "Definitions/Total Variation of Real Function"
The following 4 pages are in this category, out of 4 total.